Volume 25, Issue 6, June 1982
Index of content:

Stokes drag on a flat annular ring
View Description Hide DescriptionA surface distribution of point forces has been used to calculate the Stokes flow drag on a thin flat ring moving either parallel to or perpendicular to the axis of rotational symmetry. The results for motion parallel to the axis are confirmed by experiment. The drag is found to depend primarily upon the outside diameter of the ring and to be relatively insensitive to the inside diameter, in contrast to measurements at high Reynolds number. When the inside diameter is zero, the results agree with the known exact solutions for the circular disk.

Steady flow in a sudden expansion at high Reynolds numbers
View Description Hide DescriptionThe sudden expansion of a laminar flow in a two‐dimensional channel is examined theoretically in the limit of large Reynolds number R. Previous investigators found, from experiment and from numerical solutions of the equations of motion, that a region of closed streamlines is formed whose streamwise length is linearly related to R for R = O(10^{2}). It is desired to determine if the steady solutions to the Navier–Stokes equations continue to exhibit this relationship indefinitely for increasing R. Since solutions are sought for which the longitudinal length scale is O(R) and that in the tranvserse direction is O(1), the equations of motion reduce to the boundary‐layer equations as R→∞. These equations are solved numerically using a finite difference technique for selected values of λ, the ratio of the upstream channel half‐width to the step height. Steady solutions are found for all values of λ when the inlet velocity profile is parabolic. However, a uniform inlet velocity profile yields steady solutions with an O(R) wake length only for λ⩽λ_{ c } = 1.54. Analogous results apply in the axisymmetric case for which λ_{ c } is found to be equal 3.67.

Stability of convection rolls in the presence of a vertical magnetic field
View Description Hide DescriptionThe stability of two‐dimensional convection rolls in a horizontal fluid layer heated from below with rigid boundaries is investigated theoretically for the case when a vertical magnetic field permeates the layer. It is assumed that the magnetic diffusivity is large compared with the thermal and viscous diffusivities of the fluid. Low Prandtl numbers are emphasized in order to make the analysis directly applicable to experiments on convection in mercury and liquid sodium. The theory predicts that the onset of the oscillatory instability of convection rolls is delayed by the effects of the magnetic field and that the parameter range over which steady rolls are stable increases with the magnetic field strength. In the parameter regime investigated, there does not seem to exist a new mode of instability introduced by magnetic forces. It is also found that the magnetic field tends to increase the efficiency of the convective heat transport thereby conpensating in part for the delay in the onset of convection caused by the stabilizing effect on the Lorentz force.

Boundary conditons on the envelope function of convective rolls close to onset
View Description Hide DescriptionBoundary conditions are derived for the envelope function of convective rolls approaching a rigid sidewall at an arbitrary orientation. This generalizes previous results for parallel and perpendicular rolls. It provides the first step in a study of the convective pattern to be expected close to onset in Rayleigh–Benard cells of any cross section, and allows consideration of more complicated patterns in rectangular or circular cross sections.

Viscous decay of long internal solitary waves
View Description Hide DescriptionA formula for the viscous decay of a long internal solitary wave, propagating into a quiescent fluid in a two‐layer model was derived. The result is analogous to Keulegan’s (1948) formula for the viscous decay of long surface waves. The requirement of quiescent fluid ahead of the wave is important, as indicated by a comparison with experiments.

Weakly nonlinear steady gravity‐capillary waves
View Description Hide DescriptionThe phenomenon of three‐dimensional capillary gravity waves on deep water is considered. The solution is obtained by bifurcation from a two‐dimensional wave into three‐dimensional waves composed of the two‐dimensional wave plus a pair of oblique waves. The streamwise period of the three‐dimensional waves is doubled, and there is another period in the spanwise direction. The speed and the stability of the three‐dimensional waves are discussed.

Use of a quadrant analysis technique to identify coherent structures in a turbulent boundary layer
View Description Hide DescriptionA quadrant analysis technique based on instantaneous products u v, uϑ, and vϑ (u, v are the longitudinal and normal velocity fluctuations, ϑ is the temperature fluctuation) is used to obtain averages, in ejection and sweep quadrants, associated with coherent structures contributing significantly to Reynolds stress and heat flux in a turbulent boundary layer. Conditional averages of u, v, ϑ and products u v, uϑ, and vϑ in ejection and sweep quandrants indicate only partial qualitative agreement with those obtained by other conditional sampling techniques, with detection based on the internal front characterized by rapid changes in velocity and temperature. Although the frequency of occurrence of ejections is nearly equal to that of sweeps, only a small percentage of the total number of ejections is followed by sweeps, whereas the percentage of ejections following sweeps is negligible.

A model for the skewness of the temperature derivative in a turbulent boundary layer
View Description Hide DescriptionA simple model of temperature, based on conditional averages associated with the coherent temperature fronts, yields a distribution of the temperature derivative skewness in reasonable agreement with measurements.

Limiting particle streamline in the flow of a gas–particle mixture through an axially symmetric nozzle
View Description Hide DescriptionThe flow of gas–particle mixtures through an axially symmetric nozzle is analyzed. The particle behavior is considered under the condition where the velocity and temperature lags are small. Particular attention is paid to the particle streamlines, and the problem is solved only for the particle streamlines in the first approximation. It is rigorously proved that, when the impingement of the particles on the nozzle wall occurs in the supersonic region, there is an upper limiting point (line) in the wall region where the particles can impinge. The location of this point (line) does not depend on the particle size, but only on the nozzle geometry and the equilibrium reservoir conditions.

Spectral analysis of noisy nonlinear maps
View Description Hide DescriptionA path integral equation formalism is developed to obtain the frequency spectrum of nonlinear mappings exhibiting chaotic behavior. The one‐dimensional map, x _{ n+1} = f(x _{n}), where f is nonlinear and n is a discrete time variable, is analyzed in detail. This map is introduced as a paradigm of systems whose exact behavior is exceedingly complex, and therefore irretrievable, but which nevertheless possess smooth, well‐behaved solutions in the presence of small sources of external noise. A Boltzmann integral equation is derived for the probability distriburtion function p(x,n). This equation is linear and is therefore amenable to spectral analysis. The nonlinear dynamics in f(x) appear as transition probability matrix elements, and the presence of noise appears simply as an overall multiplicative scattering amplitude. This formalism is used to investigate the band structure of the logistic equation and to analyze the effects of external noise on both the invariant measure and the frequency spectrum of x _{ n } for several values of λ∈[0,1].

Solitary surface waves
View Description Hide DescriptionBy means of the cold electron plasma equations, we have shown that surface soliton solutions can exist in semi‐infinite plasmas.

Nonlinear ion‐acoustic waves in weak magnetic fields
View Description Hide DescriptionThe two‐dimensional dynamics of nonlinear ion‐acoustic waves in plasmas are considered. A new nonlinear equation is found which is valid for unmagnetized as well as magnetized plasmas. For exactly vanishing magnetic fields, the Kadomtsev–Petviashvili equation is recovered; for weak magnetic fields, however, the dynamics are qualitatively different from that equation, depending on amplitude. With increasing magnetic field, the new equation becomes similar (but not identical) to the Zakharov–Kuznetsov equation which was derived for very strong magnetic fields. The stability behavior of plane solitons propagating along the magnetic field is also discussed. The transition from stable to unstable behavior and some quantitative new results for the instability growth rates are presented.

Effect of low‐frequency density fluctuations on ion cyclotron waves
View Description Hide DescriptionScattering of waves in the ion cyclotron range of frequencies by low‐frequency fluctuations is analyzed in the weak turbulence approximation. Finite‐ion‐Larmor radius, multi‐ion species, ion‐drift motion, and electromagnetic terms are included in the wave kinetic equation. Conditions for significant scattering have been identified for the fast wave as well as for externally launched ion Bernstein waves. Implications for the ion cyclotron range of frequency heating in tokamak plasmas are discussed.

Wave enhancement of electron runaway rate in a collisional plasma
View Description Hide DescriptionThe effects of plasma waves on the electron runaway production rate is studied. For a wave packet with a one‐dimensional spectrum directed along the electric field and with a phase velocity range containing the critical velocity v _{ c } for runaway, the runaway production rate is found to be enhanced by many orders of magnitude. For an isotropic wave spectrum, however, the runaway production rate is reduced because of the wave‐enhanced pitch angle scattering.

Nonlinear saturation spectra of electric fields and density fluctuations in drift wave turbulence
View Description Hide DescriptionElectric field and plasma density fluctuations have been observed in the range 0.2⩽k _{⊥} ρ⩽15 in association with a long wavelength naturally occurring plasma instability. The observed spectral form and amplitude of the waves is in excellent agreement with predictions for drift waves.

Effects of curvature of the magnetic field on drift‐dissipative instability of a nonuniform plasma
View Description Hide DescriptionA study is made of the effects of curvature of the magnetic field on the drift‐dissipative instability of a nonuniform plasma. It is found that inward curvature of the static magnetic field stabilizes the drift instability produced by the particle collisions.

Energy transport equations for a bumpy torus
View Description Hide DescriptionEnergy equations are derived from kinetic theory for plasmas in a bumpy torus. It is shown that the work done by the poloidal electric field on the precession motion of the particles should not appear in the energy equation. Both finite‐ and low‐β cases are examined.

Boundary layer corrections to neoclassical ripple transport in tokamaks
View Description Hide DescriptionThe discrete nature of the toroidal magnetic field coils spoils the symmetry of the tokamak and creates small modulations of the toroidal field called ripples. Particles trapped in the ripple well drift off the flux surface and enhance both particle and heat fluxes. In the usual ripple transport calculations, the derivative of the perturbed particle distribution function in velocity space is discontinuous at the boundary between the ripple trapped and untrapped regions. To smooth out the particle distribution across the boundary, the particle distribution in the boundary layer is calculated by using the Wiener–Hopf technique. The correction to the ripple‐trapped particle distribution function away from the boundary layer and the ripple transport associated with it are obtained. It is found that the particle diffusion and heat conduction coefficients are increased by about a factor of 2 and scale like ν^{−1/2} as (ν_{eff}/ω_{ bδ})^{1/2}→1. Even for very collisionless plasmas, ν_{eff}/ω_{ bδ} = 10^{ −2}, the corrections are still non‐negligible and are about 15%.

Kinetic theory of collisionless ballooning modes
View Description Hide DescriptionA kinetic ballooning mode equation retaining full finite ion Larmor radius and ion magnetic drift resonance effects is derived by employing the high n ballooning mode formalism. It is found that the critical β is smaller than the ideal magnetohydro‐dynamic critical β, except that when η_{ i } = O(η_{ i }≡d ln T _{ i }/d ln N), that are identical. The finite Larmor radius effects reduce the growth rate but do not stabilize the mode. The ion magnetic drift resonance effects are destabilizing.

Spontaneously developing magnetic reconnections in a current‐sheet system under different sets of boundary conditions
View Description Hide DescriptionThe nonlinear development of magnetic reconnection in an isolated current‐sheet system is studied numerically. The reconnection processes develop from an initial small disturbance under different sets of typical boundary conditions. The following distinct macroscopic characteristics are shown. In a situation where plasma can freely flow out of or into the system, the fast reconnection mechanism by which the stored magnetic energy can explosively be released is set up. The magnitude of the initial disturbance influences the onset time of the fast reconnection but has little effect on the established configuration of fast reconnection. Also, even in the presence of impermeable walls placed at the top and bottom boundaries, the fast reconnection is eventually available, so that the fast reconnection process is regarded as a gross instability inherent to the system itself. On the other hand, in the tearing‐type field geometry with impermeable walls, plasma confinement in magnetic islands prevents the fast reconnection from taking place and finally leads to an oscillation on the nonlinear saturation level.